Including glutamine in a resource allocation model of energy metabolism in cancer and yeast cells

Energy metabolism is crucial for all living cells, especially during fast growth or stress scenarios. Many cancer and activated immune cells (Warburg effect) or yeasts (Crabtree effect) mostly rely on aerobic glucose fermentation leading to lactate or ethanol, respectively, to generate ATP. In recent years, several mathematical models have been proposed to explain the Warburg effect on theoretical grounds. Besides glucose, glutamine is a very important substrate for eukaryotic cells—not only for biosynthesis, but also for energy metabolism. Here, we present a minimal constraint-based stoichiometric model for explaining both the classical Warburg effect and the experimentally observed respirofermentation of glutamine (WarburQ effect). We consider glucose and glutamine respiration as well as the respective fermentation pathways. Our resource allocation model calculates the ATP production rate, taking into account enzyme masses and, therefore, pathway costs. While our calculation predicts glucose fermentation to be a superior energy-generating pathway in human cells, different enzyme characteristics in yeasts reduce this advantage, in some cases to such an extent that glucose respiration is preferred. The latter is observed for the fungal pathogen Candida albicans, which is a known Crabtree-negative yeast. Further, optimization results show that glutamine is a valuable energy source and important substrate under glucose limitation, in addition to its role as a carbon and nitrogen source of biomass in eukaryotic cells. In conclusion, our model provides insights that glutamine is an underestimated fuel for eukaryotic cells during fast growth and infection scenarios and explains well the observed parallel respirofermentation of glucose and glutamine in several cell types.

Understanding cellular metabolism is a fundamental step to develop therapies for many diseases.Especially in complex diseases like cancer or infections, where invading and colonizing cells overwhelm the host, an altered metabolism is crucial for cancer growth [1][2][3][4] and pathogenesis of microorganisms 5,6 .On the other hand, immune cells also activate their metabolism when they encounter cancer or pathogen cells 7,8 .Apart from specific disease-associated pathways, energy metabolism provides the fuel for all other processes and is a key target of cancer therapies and infection control 2,3,6,9 .
A common metabolic response of eukaryotic cells in stressful conditions or fast growth, which both require fast energy supply, is a switch from glucose respiration to fermentation or respirofermentation even under sufficient oxygen supply [10][11][12] .In human cells, this was first described by Otto Warburg in cancer cells 13 .It is worth mentioning that both glucose fermentation and respiration are important for cancer metabolism [14][15][16] .The term Warburg effect is also used for similar observations in other human cell types like activated lymphocytes and microglia cells 10,11 .A similar phenomenon in the baker's yeast Saccharomyces cerevisiae was described and named the Crabtree effect after biochemist Herbert G. Crabtree 17 .It implies a downregulation of oxidative phosphorylation.Interestingly, pathogenic yeasts like Candida albicans prefer glucose respiration (and are therefore Crabtree-negative) 18,19 , which raises questions about the evolution and trade-offs underlying this observation.It was found that C. albicans mainly uses glucose fermentation and respiration during the lag phase and exponential growth phase, respectively 20 , which indicates that fast growth of this fungus requires respiration, in contrast to the Crabtree-positive S. cerevisiae and cancer cells.This is also supported by the observed reduction in growth rate (20 to 25%) when respiration via the electron transport chain (ETC) was eliminated 21 .It is worth noting that the ETC is not the only way for C. albicans to respire (albeit the most important), the second being a shortcut of the ETC from ubiquinone to oxygen via alternative oxidases.There are different opinions in the literature on whether a reduction in respiration in C. albicans stimulates the transition from yeast to hyphae [22][23][24][25][26][27][28] .It seems that the answer strongly depends on experimental conditions.As this effect is not very clear, we will not consider it in our minimal models.
In Table 1 we provide an overview of selected eukaryotic cells and their characteristics, requirements and observed metabolic modes.It exemplifies that the observed energy metabolism is linked to growth, energy or biomass demands, enzyme characteristics, and substrate availability.Although we consider neither macrophages nor muscle cells in our study, we have included them in the table to extend the overview.
Next to glucose, the uptake of glutamine is frequently observed in cells with high energy or biomass demand [29][30][31][32] .In addition to its function as a nitrogen source, glutamine can be converted to pyruvate in eukaryotic cells in a process termed glutaminolysis in analogy to glycolysis 33,34 .The process of glutamine respirofermentation, which was named WarburQ effect in analogy to the respirofermentation of glucose 35 , was primarily described for tumor cells and can drive biosynthesis and energy generation for cellular growth 34,36 .Tumor cells, immune cells but also microorganisms entering the bloodstream are supplied with high concentrations of glucose (3.5-5.5 mM) 37 and glutamine (2.5 mM), the most abundant amino acid in the plasma 38 .It is worth noticing that, unlike S. cerevisiae, C. albicans can grow using amino acids as a sole carbon source 20 .
Since eukaryotic cells, especially in the human body, have multiple options regarding carbon sources and metabolic routes, many approaches were used to understand the evolution and trade-offs leading to the differences in energy metabolism among cells [39][40][41][42] .In the light of evolution, the concept of optimality is powerful in explaining the properties of living organisms [43][44][45] .However, standard Flux Balance Analysis (FBA) assuming yield maximization fails in elucidating the Warburg and Crabtree effects because respiration (allowing a higher ATP yield) rather than fermentation would be obtained unless an upper bound on oxygen consumption is included as an additional side constraint 46,47 .It is worth noting that the Warburg effect occurs even if sufficient oxygen is available 48 .
In contrast, to properly describe the Warburg and WarburQ effects, the reallocation of protein among enzymes is of importance.This can be modeled by resource allocation models, which use a constraint for a linear combination of velocities.By this extension to standard FBA, one can obtain results in agreement with the experimental data 10,41,42,[49][50][51][52][53] .A particular approach is called FBA with macromolecular crowding (FBAwmc); the aforementioned constraint then concerns the crowding of enzymes in the intracellular space 41 or within membranes 54,55 .Ranging from minimal models 51 to genome-scale models 42 , these computational approaches lead to the insight that the Warburg effect can be understood in terms of rate maximization or enzyme cost minimization.In contrast, respiration provides maximal yield (rather than rate) of ATP per mole of glucose, which may not be accompanied by fast growth.While a few models of glutamine respiration and/or fermentation have been proposed 35,42,56 , most models of the Warburg effect focus on glucose utilization (see references above).
Here, we propose a minimal model for the fermentation and/or respiration of glucose and/or glutamine.It is an extension of an earlier minimal model describing the Warburg effect 51 , in which only glucose was considered as a substrate.Throughout the manuscript, we will use the terminology of reactions as shown in Fig. 1 and refer to the catabolism of glucose or glutamine leading to pyruvate as glycolysis or glutaminolysis, respectively.We compare the results for human cells with those for fungal cells to understand the different cellular behaviors of eukaryotes during energy-demanding scenarios like rapid growth or infection.

Linear optimization reflects observed energy metabolism
The linear optimization problem described in the Methods section (see below) can be solved by calculating the objective function values at the outer vertices of the feasible region.Without a constraint on uptake rates, these vertices represent elementary flux modes 57 .In our model we have four elementary modes, which describe pure glutamine fermentation or respiration and pure glucose fermentation or respiration (see Figs. 1B, 2B-E and Table 2).Each of them only uses two fluxes from v 1−4 and do not include the other two.
Since the molecular weight of enzymes reflects a good estimator for enzyme cost 45,58 and is easily calculable, we carefully collected the enzyme mass per active site for each enzyme along the reactions v 1−6 (see Supplementary information).The general solution based on the parameters is given in Table 2 and shows that the optimal metabolic route is dependent on the cost of the corresponding enzymes (reflected by weights α 1−6 ) and the ATP stoichiometry (reflected by coefficients m 1−6 ).For describing the Warburg and WarburQ effects, the relative flux distributions between fermentation and respiration rather than absolute values are of importance.Therefore, the total capacity C can be normalized to be one.
The required enzyme mass per active site and reaction step is around 50−100 kDa for most cytosolic enzymes and considerably higher for enzyme complexes of the ETC, notably around 200 kDa per active site and reaction step.Since pyruvate fermentation requires only one (to lactate) or two (to ethanol) reaction steps, respiration of glucose or glutamine requires much more enzyme mass (see Fig. 3A).While in general, yeast enzymes have similar (slightly smaller) molecular weights  44,82,83 originating from proliferating progenitors 84 ideal conditions: 1.5 h 85 , barley malt: 5 h 86 ideal conditions: 1 h 87 , nonproliferative inside macrophages 44 Process requiring additional biomass or energy cytokine and ROS production upon activation 83 muscle contraction 88 hyphae formation 18,89 Glucose availability via blood, 3.5−5.5 mM 37 sugar >250 mM during ethanol production not quantified in phagolysosomes; via blood during invasive growth Glutamine availability via blood, 2.5 mM 38 not a primary source during ethanol production not quantified in phagolysosomes; via blood during invasive growth

Observed energy metabolism
Warburg and WarburQ effects 34,36 Warburg effect upon activation 90 aerobic glycolysis during heavy exercise 88 Crabtree effect 17 Crabtree-negative 19 as their human homologs, the ETC complexes of yeast are significantly smaller due to non-homologous enzymes like complex I substituents (see Fig. 3A and cf.Supplementary information).In contrast, glutamine assimilation in yeast requires an alternative pathway, namely the GS-GOGAT cycle 59 , with higher enzyme masses than the human variant.All of these factors, together with the different ATP yield per NADH, influence the optimal metabolic route to maximize the ATP rate per enzyme mass.In both human and S. cerevisiae cells, glucose fermentation is superior over glutamine as a substrate and over respiration-as a metabolic mode (see Fig. 3B).The superiority of glucose fermentation over glucose respiration is the classical Warburg effect.Due to the smaller enzymes of the ETC, the difference between glucose fermentation and other metabolic routes is smaller in the yeast models than in the human cell model.Interestingly, the usage of a high-yield complex I (as in the Crabtree-negative C. albicans) instead of the low-yield complex I analog (as in S. cerevisiae) leads to glucose respiration as the optimal metabolic route for maximal ATP generation (see Fig. 3B).Accordingly, the optimal solutions for the human cancer model and Candida model are reached at the blue dot and blue circle, respectively, in Fig. 2A.Summarizing the above, the model reflects well the metabolic behavior of the eukaryotic cells and confirms glucose to be the primary source for energy metabolism.However, glutamine respiration, as well as fermentation, generates comparable amounts of ATP, especially in the Candida model, and can substitute glucose under limiting scenarios as elaborated in the next section (see Fig. 3B).

Flux constraints cause mixed energy metabolism
In many situations, fluxes through the reactions of energy metabolism in human cancer (or immune) cells and yeast cells are limited.The supply of the carbon sources glucose (v 1 ) or glutamine (v 4 ) is high but certainly limited for cancer cells or pathogenic yeasts like C. albicans.When C. albicans cells are engulfed by macrophages, their glucose supply is limited, so that they switch to fatty acid consumption.Inside non-vascularized tumors, nutrients and oxygen are scarce.Fluxes through fermentation or respiration (reactions 2 and 3, respectively) may reach their maximal capacity due to the lack of sufficient enzymes, cofactors, or oxygen.We simulate this by maximal velocity constraints and a recalculation of the optimal flux distribution in the energy metabolism based on our simplistic model.Since human (cancer) cells and S. cerevisiae show a preference for glucose fermentation 48,60 , we analyzed the influence of reduced capacities on the fluxes through the glycolysis and fermentation reactions 1 and 2 (Fig. 4).In contrast, in Candida-like species with a high-yield complex I, glucose respiration showed the highest ATP flux.Hence, for these species, we simulate a capacity limit on fluxes through the lumped glycolysis and respiration reactions 1 and 3.
A reduced glucose supply or fermentation capacity leads, in both human and yeast cells, to a mixed respirofermentation as the optimal energy metabolism, where the fraction of respiration is determined by the flux capacity constraints that limit v 1 (e.g., the capacity of glucose transporters) and v 2 and the constraint on enzyme costs (see Fig. 4A, B).Further, the usage of glutamine is affected by the constraint on glucose supply (v 1 ), but not by a reduced fermentation capacity (v 2 , see Fig. 4D, E).This means that, in the case of human and S. cerevisiae models, reducing fermentation does not force cells to use glutamine (see also Fig. 5A).In other words, the importance of glutamine uptake for those cells is not confirmed by our model of energy metabolism; it is suggested that glutamine more likely serves biomass synthesis requirements.
In the Candida-like cells with a preference for glucose respiration, the influence of capacity constraints along the metabolic route (v 1 , v 3 ) is different from the one in human and S. cerevisiae cells.The fraction of respiration appears to decrease when respiratory capacity is lowered under a certain threshold, but shows also a nonlinear and step-wise influence by glucose supply limitation, favoring more fermentation (see Fig. 4C).In a similar way, glutamine usage increases as glucose supply is more and more limited; the same behavior is observed in case of the limitation of respiration provided that glucose is also limited (see Fig. 4F), indicating a more pervasive importance of glutamine in the energy metabolism of Candida spp.The solution space for the Candida model with the limitation for v 1 and v 3 is shown in Fig. 5B; the optimal strategy is, as mentioned before, the combination of respiration and fermentation, utilizing both glucose and glutamine (see Table 3).The solution for baker's yeast with the limitation of v 1 is respirofermentation as well, yet without glutamine consumption (v 4 = 0, Fig. 5A and Table 3).

Discussion
Many experimental and theoretical studies investigated the Warburg and Crabtree effects.There is evidence that many cancer hallmarks are the results of the Warburg effect 61 .Recently, the utilization of glutamine by cancer cells has aroused more and more attention 34,36,62 .While glutaminolysis is extensively studied experimentally, conclusive theoretical models for explaining the utilization of glutamine in terms of fast and efficient growth of cancer cells are still scarce.
To reach a better understanding of the energy metabolism in human cells, we extended a minimal model of the Warburg effect to study, in addition, the usage of glutamine via fermentation, which is related to the WarburQ effect, and via respiration.This led to a minimal model of this effect, since it involves six reactions only.In contrast, earlier models of this effect involve more than 70 reactions 35,56 or are even genome-scale 42 .Like in the earlier three-reaction model 51 , the inclusion of an upper limit on the uptake of glucose allows us to model the respirofermentation of this substrate (Warburg and Crabtree effects).It is worth mentioning that Otto Warburg himself described a mixture of glucose fermentation and respiration 48 .
In our model, the reaction rates are considered as the variables of the optimization problem, without making any assumptions regarding rate laws.The rates are considered to be proportional to the enzyme concentrations, so that the side constraints can be written in terms of the rates.In more detailed approaches, the interrelation between the rates due to utilization of common metabolites such as pyruvate was taken into account 50,63 .Since enzyme kinetic rate laws are nonlinear, the side constraints are nonlinear in terms of metabolite concentrations even if they are linear in terms of enzyme concentrations.It was shown that the optimal solutions in any metabolic network correspond to elementary modes 50,63 , like in our simplified approach.
By parameterization we adapted our linear model also to S. cerevisiae and fungal pathogens like C. albicans to understand the role of glutamine in Fig. 3 | Cross-species comparison of the particular routes of energy metabolism.Colors indicate the substrates of the lumped reactions (blue, glucose; red, glutamine; empty circles, respiration; full circles, fermentation).A For each lumped reaction, the sums of the molecular weights per catalytic center of all enzymes are visualized and allow a comparison of the related costs in the human, baker's yeast, and C. albicans models.B The costs from (A) are used to calculate the ATP rate per MDa of molecular enzyme mass for the four possible elementary modes.Note that not all of the fluxes need to be active in the optimal state.Parameter values: Supplementary Tables 1, 2; C = 1.

Metabolic route
Coordinates (v 1 , v 2 , v 3 ) Objective value Glutamine respiration 0; 0; Glucose fermentation The cost parameter C is normalized to be unity.For simplicity's sake, the subscript ATP in the stoichiometric coefficients was omitted.In A-C, the optimal flux allocation between v 2 (fermentation) and v 3 is shown as v 3 /(v 2 + v 3 ), in %, in human, S. cerevisiae, and Candida-like models, respectively.Analogously, D-F show the optimal flux allocation between v 1 (glucose acquisition) and v 4 (glutamine acquisition) as v 4 /(v 1 + v 4 ), in %, in the corresponding models.For human and S. cerevisiae models, the optimal flux allocations are shown as functions of maximal fluxes through reactions 1 and 2 (v 1cap , v 2cap ); for the Candida-like model, as functions of maximal fluxes through reactions 1 and 3 (v 1cap , v 3cap ).comparison to glucose in energy metabolism of other eukaryotes and infection scenarios.As it was mentioned above, C. albicans can grow solely on amino acids and has access during the bloodstream infection to both glucose and glutamine.Baker's yeast can utilize glutamine as a (sole) nitrogen source 64,65 , but not as a sole carbon source 66 .Moreover, S. cerevisiae cannot use amino acids in general as a sole carbon source 66 , and does not use glutamine as a carbon source when glucose is available 65 .Glutamine fermentation possibly happens during wine fermentation, which leads to faster fermentation and promotion of volatile compound formation 67 .Glutamine, as well as another good nitrogen source, ammonia, represses many genes connected with nitrogen metabolism 65 .
Our results for human and baker's yeast models indicate and confirm glucose as the preferred substrate and fermentation as the preferred route for cost-efficient generation of energy in the form of ATP.It seems reasonable to also consider the fact that proliferating cells need carbon atoms, and it should not be rational to waste all of them as CO 2 during respiration 11,68 .Interestingly, the aforementioned strategy does hold true for human cells and S. cerevisiae but not for our model of the fungal pathogen C. albicans, where glucose respiration is superior over fermentation.This is in line with experimental findings that C. albicans is Crabtree-negative and primarily uses respiration during fast growth phases 19 .
In addition to the analysis of glutamine utilization, we also explained the observation that C. albicans prefers glucose respiration over glucose fermentation (i.e., Crabtree-negative), by the fact that this fungus has a complex I different from the one in baker's yeast and several enzymes different from those in humans with respect to their molecular masses.These differences can be captured by our minimal model.It is an interesting question why in S. cerevisiae (in comparison to C. albicans), the typical complex I is substituted by an enzyme that does not pump protons, so that the ATP yield is lower than in higher animals, for example.This can be regarded as a trade-off between a high-yield/low-rate pathway (e.g.respiration using complex I) and a low-yield/high-rate pathway (e.g., fermentation), which can even be adjusted to varying environmental conditions 39 .
Our results support the finding that glutamine is a valuable resource fueling fast tumor growth 36 .In addition to cancer metabolism, the results also have implications for infection scenarios and host-pathogen interaction.Glutamine has a high blood concentration 38 , albeit lower than glucose concentration 69 ; our optimization results indicate that under limited glucose and fermentation scenarios, glutamine is a comparably efficient substrate.Further, our results show that glutamine respiration and glutamine fermentation have a comparable ATP rate per enzyme mass, which is, moreover, nearly identical to the one of glucose respiration.While it is inferior to glucose fermentation in the human and baker's yeast model, glutamine respiration even surpasses glucose fermentation in the Candidalike model.This supports the experimental finding and the view that, in cancer cells, glucose is fermented to lactate to generate energy, while glutamine respiration is used in addition to fuel the growth of cancer cells and biomass generation 70 .
Our results show that when no constraint on substrate uptake is included, the optimal flux distributions coincide with elementary modes, in agreement with earlier theoretical findings 40,50,63 .In contrast, when glucose uptake is considered to be significantly limited, a superposition of fermentation and respiration is obtained.
In future studies, it is worthwhile considering the excretion of fermentation products such as lactate to be reversible, as it was done in an earlier model 53 .To achieve that, a combination of that model and the one proposed in this article could be constructed.A further interesting extension is to consider the overall reaction from glutamine to pyruvate to be reversible.This would allow one to shed light on the energy balance of nitrogen storage and incorporation of glutamine into proteins.Another interesting point is that, for our model, the molecular masses of the enzymes were sufficient to predict a pathway's cost, while other studies employ more elaborated formulations of pathway cost functions 42,45,58,71 .It is worth investigating whether this result of ours can be generalized.Such a suggestion could be corroborated by a study of the Warburg effect analog in Escherichia coli.In an earlier study of this so-called overflow metabolism, it was also pointed out that the preference of fermentation over respiration is a result of the latter pathway's higher costs 72 .
In conclusion, our model is in good agreement with experimental observations of energy metabolism in eukaryotic cells.It shows that glutamine is an underestimated substrate for energy metabolism in comparison to glucose.Also it is versatile since it has a comparable ATP production rate per enzyme mass via respiration and fermentation and serves as a carbon as well as nitrogen source for biomass.The cross-species comparison disclosed that the optimal resource allocation in these metabolic models is influenced by the cost of the underlying pathways and is not uniform across the kingdoms of life.Our model can be applied by parameterization and stoichiometric adjustments to other species, including pathogens.The understanding of metabolic resource allocation is a crucial step for the identification of drug targets against microbial pathogens or cancer cells.

A minimal model to describe the Warburg and WarburQ effects
To achieve a model as small as possible, we lump entire pathways such as the tricarboxylic acid (TCA) cycle and oxidative phosphorylation into overall reactions (see Fig. 1), to describe the metabolic choices of a cell.We formalize a linear optimization problem of ATP generation under limited resources for enzyme synthesis.Reaction 1 represents the uptake of glucose as well as the upper part of glycolysis and reaction 4 represents the uptake of glutamine and its conversion to pyruvate (cf.Fig. 1A, B).Pyruvate is chosen as a central hub since it represents a branching point between fermentation (reaction 2, products like lactate or ethanol) and respiration (reaction 3, oxidation to CO 2 ), and it is the only internal metabolite in this minimal model.Due to the addition of reaction 4 in comparison to earlier models, we explicitly consider the energy-rich cofactors NADH and FADH 2 as external metabolites, which contribute to energy generation via the electron transport chain and ATPase by the conversion of ADP into ATP (reactions 5 and 6).v 1 -v 6 denote the steady-state fluxes through the six aforementioned reactions.Like in previous models, we formulate ATP production rate as an objective function at a steady state, with the stoichiometric coefficients m of cofactor generation listed in Table 4.Only the stoichiometric coefficients m ATP shape the objective function; the contribution of NADH and FADH 2 to the energy balance is taken into account by considering the ETC in the model.
In human cells, complex I of the ETC comprises 45 subunits and forms a supercomplex with other respiratory chain complexes, namely III and IV 73,74 .While in S. cerevisiae, the typical complex I is substituted by a smaller enzyme, encoded by the gene NDI1, which does not pump protons, Candida albicans and other pathogenic yeasts have both complex I and NDI1 analogs; this affects the ATP yield per NADH 75 .Accordingly, the three organisms show different ATP-over-glucose yields of respiration, notably 32 in C. albicans 20 , 16-18 in S. cerevisiae 12,76 , and about 30 in (healthy) humans 77 .Due to the immense difference between metabolisms of different cancer cell types, ATP yield in certain cancer cells may vary, but is assumed to be comparable to the value in healthy cells.
To account for the differences in ATP production, we compare three models, namely human cancer cells, S. cerevisiae with low-yield ETC and a Candida-like model with high-yield ETC and baker's yeast enzyme masses (except for complex I).For humans and C. albicans, the following stoichiometric calculations can be done: one molecule of NADH starts a chain of reactions (beginning with complex I) that pump 10 H + , out of which 4 H + are pumped by complex I itself 75,78 .4 H + are needed to produce 1 ATP,  4).Since NDI1, which is S. cerevisiae's substituent of complex I, does not pump protons, only 6 H + are produced per NADH, so the yield is 1.5 ATP per NADH (Table 4) 75 .
As in previous models of the Warburg effect, we further assume a limit on the total flux sum (C), and the costs of fluxes are weighted with respect to their enzyme costs (α).By normalization, we set C = 1 and calculate α i as enzyme mass (in MDa) per catalytic site across all reactions of the pathway (see Supplementary Tables 1 and 2).As a result, J ATP can be interpreted as the amount of ATP per mole of substrate and per MDa of molecular enzyme mass required to catalyze the reactions.We used enzyme mass (molar mass based on amino acid sequence, as listed in UniProt 79 ) as an estimation since it is available for enzymes with known amino acid sequence and has proven to be a good approximation of synthesis cost 45,58 .Further, the number of catalytic sites was inferred from PDB structures 80 and published experimental studies (see Supplementary Tables 1, 2) to correctly account for the involved large enzyme complexes.The resulting enzyme masses per catalytic site and the corresponding cost factors α i are listed in Table 5.
Our focus is on comparing glucose and glutamine as possible substrates rather than lactate or other fermentation products, which could be reabsorbed.For simplicity's sake, we thus consider all reactions to be irreversible, so that the possible re-uptake of fermentation products 53 is neglected.
In addition to the steady-state and cost constraints, we introduce a flux constraint in extension to the basic model.By modeling capacity constraints of certain reaction fluxes, we can mimic and study effects like the limited uptake rate of glucose or a saturation of enzymes like lactate dehydrogenase.
Overall, the resource allocation problem for the system under study (see Fig. 1) can be formally written as: Maximize ð ATP production rateÞ J ATP ðvÞ ¼ where v i is the flux (with the dimension [mM/s]) through the corresponding reaction i; m ATP i is defined by the coefficients of ATP generation of the lumped reactions as described in Table 4 (first row).
In this formulation, a flux distribution of v 1−6 is calculated, which maximizes the ATP production rate according to the mass stoichiometry coefficients m depicted in Fig. 1 and Table 4.This linear maximization of ATP production (equation ( 1)) is constrained by a steady-state assumption for internal metabolites (including energy-rich cofactors, equation ( 2)).Due to the required mass balance in a steady state, the system has three degrees of freedom, and the solution space can be visualized in a plot of v 1−3 (Figs.2A, 5), v 4 being unambiguously defined by the rest of the carbon-transporting fluxes.
The sum of the fluxes v i weighted by the protein mass coefficients α were limited by a cost constraint C, which is normalized to be unity (inequality (3)).Inequality (4) expresses our assumption that all reactions are irreversible.These constraints limit the feasible region to a closed polyhedron (see Fig. 2A).The last inequality ( 5) is taken into account only in an advanced model variant, where we include, in addition, an upper bound on the uptake rates of glucose and/or glutamine (see Figs. 4, 5).
Based on the human enzymes, homologous enzymes in S. cerevisiae were determined and differing reactions and non-homologous enzyme complexes are considered.Since enzyme masses of homologs are mostly comparable across eukaryotes, we simulate a Candida-like organism by using the enzyme masses from S. cerevisiae except the complex I, which is not present in this organism 73 .

Model implementation
The linear optimization problem phrased in equations ( 1)-( 5) was solved by the MATLAB routine linprog.The objective function was formulated as ATP production rate (1) divided by the cost (3).To simulate the optimal flux distribution for situations with additional constraints (5), a mesh of varying constraint values was generated in MATLAB, and linear optimization was performed for every node.The corresponding code can be found at https:// git.uni-jena.de/qe45cow/warburq-minimal-model.

Reporting summary
Further information on research design is available in the Nature Research Reporting Summary linked to this article.

Fig.
Fig.1| Metabolic routes and simplified model of energy metabolism.A The substrates glucose (Glc) and glutamine (Gln) can both be respired to CO 2 or fermented to lactate (Lac) in humans or ethanol (Eth) in S. cerevisiae and C. albicans19,91 via different routes (colored and shaded boxes).B Pyruvate (Pyr) is used in our model as a hub between the different metabolic routes.Overall, reactions 1-4 describe lumped pathways which generate energy via ATP and the cofactors NADH or FADH 2 .The conversion of NADH or FADH 2 to ATP is described by reactions 5,6.v 1v 6 , fluxes through reactions 1-6 at steady state.

Fig. 4 |
Fig.4| Heat map showing the optimal flux allocations for different scenarios.In A-C, the optimal flux allocation between v 2 (fermentation) and v 3 is shown as v 3 /(v 2 + v 3 ), in %, in human, S. cerevisiae, and Candida-like models, respectively.Analogously, D-F show the optimal flux allocation between v 1 (glucose acquisition) and v 4 (glutamine acquisition) as v 4 /(v 1 + v 4 ), in %, in the

Fig. 5 |
Fig.5| Solution spaces spanned by the fluxes through the model's reactions, analogous to the one presented in Fig.2A, for two other models.Due to the steady-state condition, the solution space is uniquely described by the fluxes v 1−3 .A S. cerivisiae model and the case of limitation of v 1 .The flux v 4 , which is not shown in the plot, is equal to zero in the optimal state -therefore, the optimal strategy for this model is glucose respirofermentation.B Candida model and the case of limitation of v 1 and v 3 .The flux v 4 (not shown) is positive in the optimal state.Therefore the optimal solution is the respirofermentation of both glucose and glutamine.Faces of the initial polyhedron -red (A), yellow (B); blue -face/s caused by the limitation of v 1 (A), v 1 and v 3 (B); green dot, solution of the optimization problem.C, E Projection of the polyhedron onto the v 1 -v 3 plane.D, F Projection of the polyhedron onto the v 2 -v 3 plane.

Table 1 |
Overview of selected eukaryotic cells and their characteristic features of energy metabolism 1 | Metabolic routes and simplified model of energy metabolism.A The substrates glucose (Glc) and glutamine (Gln) can both be respired to CO 2 or fermented to lactate (Lac) in humans or ethanol (Eth) in S. cerevisiae and C. albicans 19,91 via different routes (colored and shaded boxes).B Pyruvate (Pyr) is used in our model as a hub between the different metabolic routes.Overall, reactions 1-4 describe lumped pathways which generate energy via ATP and the cofactors NADH or FADH 2 .The conversion of NADH or FADH 2 to ATP is described by reactions 5,6.v 1v 6 , fluxes through reactions 1-6 at steady state.
Fig. 2 | Solutions of linear optimization model of energy metabolism including glutamine.A Solution space spanned by the fluxes through the model's reactions, with the parameter values for a human cell model.Due to the steady-state condition, the solution space is uniquely described by the fluxes v 1-3 .All constraints form the polyhedron of the solution space (gray), and pure metabolic routes are represented by colored vertices: glucose (GLC) respiration or fermentation (open or filled blue circle, respectively) as well as glutamine (GLN) respiration or fermentation (open or filled red circle, respectively).Coordinates v 1 , v 2 , v 3 of the vertices in the order given above [0.0707,0,0.1414],[1.7804, 3.5607, 0], [0, 0, 0.0972], [0, 0.2863, 0]; see also Table2.Note that the origin of the coordinates is located at the rear bottom right corner.B-E Elementary modes in our model that correspond to the polyhedron's vertices.

Table 3 |
Flux values of the optimal solutions shown in Fig.5

Table 5 |
Total enzyme mass per catalytic site of each overall reactionFor Candida-like species, a high-yield complex I of C. albicans is used instead of the low-yield but small S. cerevisiae variant. 1, glucose degradation (up to pyruvate); 2, pyruvate fermentation; 3, pyruvate respiration; 4, glutamine degradation (up to pyruvate); 5 and 6, reactions of ETC.